Abstract
Stability regions of nonlinear dynamical systems may suffer drastic changes as a consequence of parameter variation. These changes are triggered by local or global bifurcations of the vector field. In this chapter, these changes are studied for two types of local bifurcations on the stability boundary: saddle-node bifurcations and Hopf bifurcations on the stability boundary. Local and global characterizations of the stability boundary (the topological boundary of stability region) will be developed at the bifurcation points and the behavior (changes) of stability boundaries and stability regions at these bifurcations will be studied.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Amaral, F.M., Alberto, L.F.C., Gouveia, J.R.R.: Stability boundary and saddle-node equilibrium points on the stability boundary of nonlinear autonomous dynamical systems. Dyn. Syst. 01, 1–23 (2017)
Amaral, F.M., Alberto, L.F.C.: Bifurcações Sela-Nó da Região de Estabilidade de Sistemas Dinâmicos Autônomos Não Lineares. Trends Appl. Comput. Math. 1, 71–80 (2016)
Amaral, F.M., Alberto, L.F.C.: Stability boundary characterization of nonlinear autonomous dynamical systems in the presence of saddle-node equilibrium points. Trends Appl. Comput. Math. 2, 143–154 (2012)
Amaral, F.M., Alberto, L.F.C.: Stability region bifurcations of nonlinear autonomous dynamical systems: type-zero saddle-node bifurcations. Int. J. Robust Nonlinear Control 21, 591–612 (2011)
Amaral, F.M., Alberto, L.F.C.: Type-zero saddle-node bifurcations and stability region estimation of nonlinear autonomous dynamical systems. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 22(1), 1250020–1 (2012)
Chiang, H.D., Hirsch, M.W., Wu, Felix, F.: Stability region of nonlinear autonomous dynamical systems. IEEE Trans. Autom. Control 33, 16–27 (1988)
Chiang, H.-D., Wu, F., Varaiya, P.: Foundations of direct methods for power system transient stability analysis. Circ. Syst. 34, 160–173 (1987)
Chiang, H.-D: Alberto, L.F.C.: Stability Regions of Nonlinear Dynamical Systems, Theory, Estimation and Applications. Cambridge University Press (2015)
Gouveia, J.R.R., Amaral, F.M., Alberto, L.F.C.: Stability boundary characterization of nonlinear autonomous dynamical systems in the presence of a supercritical Hopf equilibrium point. Int. J. Bifurcat. Chaos 23(12), 1350196 (2013)
Gouveia, J.R.R., Amaral, F.M., Alberto, L.F.C.: Subcritical hopf equilibrium points in the boundary of the stability region. TEMA São Carlos 17(2), 211–224 (2016)
Gouveia, R.R.: Stability region bifurcations induced by local bifurcations of the type Hopf. Thesis - EESC, University of Sao Paulo (2015)
Guedes, R.B, Alberto, L.F.C., Bretas, N.G.: Power system low-voltage solutions using an auxiliary gradient system for voltage collapse purposes. Power Syst. 20, 1528–1537 (2005)
Gunther, N., Hoffmann, G.W.: Qualitative dynamics of a network model of regulation of the immune system: a rationale for the IgM to IgG switch. J. Theor. Biol. 94(4), 815–855 (1982)
Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Bull. Am. Math. Soc. 76(9), 1015–101 (1970)
Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Springer, New York (1995)
Palis, J.: On morse-smale dynamical systems. Topology 8, 385–405
Sotomayor, J.: Generic bifurcations of dynamical systems. Dyn. Syst. 549 (1973)
Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1989)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Alberto, L.F.C., Amaral, F.M., Gouveia Jr., J.R.R. (2018). Bifurcations and Stability Regions of Nonlinear Dynamical Systems. In: Edelman, M., Macau, E., Sanjuan, M. (eds) Chaotic, Fractional, and Complex Dynamics: New Insights and Perspectives. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-68109-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-68109-2_7
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-68108-5
Online ISBN: 978-3-319-68109-2
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)